3.11.5 Failure Modes & Desk Practice

Basic Prerequisites: 03 · Calibrating Local Vol and 04 · Calibrating Stochastic Vol.

---

1. Intuition & Practical Objective

This page names the three ways calibration goes wrong - overfitting, instability, non-identifiability - ties each to a first-principles cause, and then describes how a real desk survives them (market data, conventions, and the discipline of distrusting the calibrated model). The objective is not cynicism: it is knowing precisely which assumptions to distrust so the residual model risk can be measured and managed.

Three structural facts do the damage:

1. Differentiation amplifies noise. Local vol needs second derivatives of prices; noisy quotes ⇒ huge local-vol spikes or negative variance. (Ill-posedness.)
2. The smile fixes combinations, not parameters. SABR’s $\beta /\rho$, Heston’s $\kappa /\eta$ are collinear; different parameters fit identically. (Non-identifiability.)
3. The model is static but the market is not. A parameter set fit to today is not a law of motion; recalibration drift is real P&L risk. (Overfitting + model risk.)

---

2. Mathematical Ground Truth & Derivations

Ill-posedness of the local-vol inversion (Gatheral 1.10; Bergomi 2.19). The local variance is

$$v_L(y,T)=\frac{\partial w/\partial T}{1-\frac{y}{w}{w}_y+\frac{1}{4}(-\frac{1}{4}-\frac{1}{w}+\frac{y^2}{w^2})w_y^2+\frac{1}{2}{w}_{yy}},$$

which contains $w_{yy}$ - a second finite difference. If observed with noise $\epsilon$ at grid spacing $dy$, the second difference has error $\sim \epsilon /d{y}^2$, and the denominator $\frac{1}{2}{w}_{yy}$ can be driven to $0$ (or negative, i.e. strike arbitrage / negative butterfly). Near a zero denominator, $v_L\to \pm \infty$. This is the textbook ill-posed inverse problem: the forward map (smile → local vol) is smoothing, so its inverse is unstable.

Non-identifiability (Bergomi Ch 6, Ch 8; Gatheral Ch 3). The ATMF skew is, to first order, a weighted integral of the instantaneous spot/vol covariance:

$${\mathcal{S}}_T=\frac{1}{{\hat{\sigma }}_T^{2}T}{\int }_0^T\frac{T-t}{T}⟨d\ln S_{t}d{\hat{\sigma }}_T(t)⟩dt\text{(Bergomi eq 2.89)}.$$

The combination is fixed by the smile; how it is split between $\rho$ and $\eta$ (or $\beta$ and $\rho$) is not. That split changes the dynamics ($R_T$, the skew-stickiness ratio) without changing the static fit - hence identical fits with different hedge ratios.

Conventions that quietly change the answer. Calibration output is only meaningful given a convention set (Bergomi Ch 5 on variance swaps makes this vivid): implied vs realized vol, log-returns vs arithmetic returns in the variance payoff (${\ln }^2$ vs standard returns flips the sign of the third-order term), $\ln K/F$ vs $\ln K/S$ moneyness, per-point vs per-unit vol scaling, the dividend model (continuous yield vs discrete), and the discount curve. Two desks with the same data and model but different conventions report different calibrated parameters.

---

3. Computational Implementation - instability in numbers

Reconstruct local vol from a smooth smile with and without realistic bid/ask noise (~0.4 vol pts, white across strikes). The differentiation amplifies the noise catastrophically. Stdlib only.

A 0.4-vol-point wiggle in the smile blows one reconstructed local variance up to $4.3\times 10^7$ - about nine orders of magnitude above the true level ($\sim 0.03$) - and swings the butterfly denominator to near zero. This is why desks never invert raw quotes: they first fit a smooth, arbitrage-free parametric surface (SVI, a parametric LV, a calibrated SV model) and only then invert, and they clip/smooth the result. The instability is structural, not a code bug.

---

4. Failure Modes & First-Principles Breakdowns (numbered)

1. Ill-posed inversion (differentiation amplifies noise ~$1/d{y}^2$). The local-vol map needs second derivatives of prices; raw-quote noise explodes it (Experiment above). Fix: smooth/parameterize the smile first, regularize, clip local variance to be positive and bounded.
2. Non-identifiability (smile fixes combinations, not parameters). SABR $\beta /\rho$ and Heston $\kappa /\eta$ trade off; identical fits, different hedges. Fix: fix a convention ($\beta =0.5$ equity, $\beta =1$ FX), use term structure, or ridge-regularize toward a prior (04 · Stochastic Vol).
3. Overfitting (rich model, sparse data). Enough parameters drive today’s residuals to zero and tomorrow’s prices to nonsense - the far wings have few quotes, so that is where overfit surfaces break. Fix: cross-validate, penalize (ridge), check no-arbitrage (butterfly $\ge 0$, total variance increasing in $T$).
4. Convention drift. Two desks with identical data and model but different conventions (moneyness, log vs arithmetic returns, dividend model, discount curve, per-point scaling) get different parameters - a silent reconciliation risk.
5. Recalibration drift (model risk). The calibrated model is static; when the market moves you recalibrate, and the change in the calibrated surface is itself unhedgeable P&L (Bergomi Ch 2.6: LV future skews are not lockable). The desk prices the deal and the cost of holding the model’s dynamics.
6. Incomplete-market assumption ($\varphi =0$). Setting the price of vol risk to zero is a choice; it is invisible in the vanilla fit but changes exotic prices.

Desk practice distilled. Fit in implied vol (the quoted unit). Smooth, then invert (never invert raw quotes). Fix or regularize the non-identifiable parameters. Check no-arbitrage after every calibration. Report residuals - they flag where the model is structurally wrong. And treat every calibration as a snapshot, not a law of motion.

---

5. References

• Gatheral, The Volatility Surface
• Bergomi, Stochastic Volatility Modeling
• Duffy, Finite Difference Methods in Financial Engineering
• Hull, Options, Futures, and Other Derivatives

---

6. Connected Graph Bridges

• Back: 04 · Calibrating Stochastic Vol · Index Hub
• Forward: 06 · Advanced Extensions
• Sibling: Heston & SABR · Implied Volatility Surfaces · Counterparty Risk & XVA